In [1]} the authors showed rigidity of the Poiseuille flow in the periodic channel for solutions close in vorticity in \(H^{5+}\)  regularity and rigidity of the Kolmogorov flow on the rectangular torus \(2_\delta =[0,2\pi \delta )\times [0,2\pi )\)  for solution near the Kolmogorov vorticity in \(H^{3+}\)  regularity. However, the situation for the Kolmogorv flow on the square torus was shown to be completely different: the authors constructed non-trivial steady solutions (non-shear and not in the kernel of the linearized Euler equations) to the Euler equations arbitrarily near the Kolmogorov flow in analytic regularity via a fixed point argument. This, in turn, implies that the linear inviscid damping for solutions near the Kolmogorov flow on 2 obtained in [2]} cannot be extended to the non-linear level, no matter the regularity of the initial perturbation.
